You don't say how the two types of leaves are disposed on the tree, but I'm guessing your tree is a cultivated variety (cultivar) grafted onto a rootstock. Those are effectively two different but closely related trees, which differ in terms of leaf (as you have noticed) and fruit, and probably other characters such as vigour and maybe soil-type preference.

The grafting may have been done low down or higher-up where the main branches start. There may be a change in girth of the trunk at the graft union. Chances are, any leaves on branches arising below the graft are the big ones. If the tree has not been maintained well, branches from the rootstock could have been allowed to persist, which could mean they are forming a substantial part of the tree now (as they would probably be more vigorous than the cultivar). In principle, the branches from the rootstock should be pruned out, but you would need to consider the effects of doing so. Many people prune cherries (and other Prunus) in winter (like many other trees). The problem with doing that with Prunus is that it lets silverleaf disease in - which can lead to the death of the tree a branch at a time. The best way to avoid silverleaf is to make only small pruning cuts (which heal quickly; so don't let unwanted branches get big) and to prune when the tree is in active growth so it can heal quickly - July. By all means take other advice on that issue; opinions do vary. (And painting pruning cuts with wound paint is reckoned not to be very effective.)

If you'd like to post a photo of the whole tree, I could make some better-tuned comments.

2. wait a couple of weeks, to give the tree a chance to reroute its sap flow somewhat (it may bleed sap during that time)

3. prune those stumps close to the trunk (hopefully it will bleed a bit less now)

To elaborate on 3:

- because of the way the trunk is engulfing the branches, it will be tricky to make the cuts without cutting the trunk somewhat as well; be as careful as you can

- use a pruning saw, not a carpentry saw; pruning saws have ground teeth not set teeth, which enables a smooth finish to the cut - which improves healing and inhibits ingress of pathogens a bit

- more for general reference than in this case, but pay attention to the branch collar, and keep your cut just to the outside of it (see http://pubs.ext.vt.edu/430/430-455/430-455.html). In your immediate case, I think the collars are lost in the engulfing of the trunk.